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Integration by Substitution

Find the inte...

Question

Find the integrals of the functions.
$\int \frac{c o s 2 x + 2 s i n^{2} x}{c o s^{2} x} d x .$

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Solution

$\int \frac{c o s 2 x + 2 s i n^{2} x}{c o s^{2} x} d x = \int \frac{1 - 2 s i n^{2} x + 2 s i n^{2} x}{c o s^{2} x} d x (∴ c o s 2 x = 1 - 2 s i n^{2} x) = \int \frac{1}{c o s^{2} x} d x = \int s e c^{2} x d x = t a n x + C$

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